Pinchasi–Sharir conjecture on planar tangent graphs

A tangent graph has one vertex for each disc and an edge when the corresponding discs are tangent. Pinchasi–Sharir tangent-incidence conjecture. (i) Planar tangent graphs with $n$ vertices have at most $n^{4/3}\operatorname{polylog}(n)$ edges. (ii) More generally, $m$ red discs and $n$ blue discs, including the special case of $n$ blue points, can touch at most $((mn)^{2/3}+m+n)\operatorname{polylog}(m,n)$ times. The source reports a best known upper bound of $n^{3/2}\log n$ and leaves the conjecture open.